The Annals of Statistics

Multi-Armed Bandits with Discount Factor Near One: The Bernoulli Case

F. P. Kelly

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Each of $n$ arms generates an infinite sequence of Bernoulli random variables. The parameters of the sequences are themselves random variables, and are independent with a common distribution satisfying a mild regularity condition. At each stage we must choose an arm to observe (or pull) based on past observations, and our aim is to maximize the expected discounted sum of the observations. In this paper it is shown that as the discount factor approaches one the optimal policy tends to the rule of least failures, defined as follows: pull the arm which has incurred the least number of failures, or if this does not define an arm uniquely select from amongst the set of arms which have incurred the least number of failures an arm with the largest number of successes.

Article information

Ann. Statist., Volume 9, Number 5 (1981), 987-1001.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 90C40: Markov and semi-Markov decision processes
Secondary: 62L05: Sequential design 62L15: Optimal stopping [See also 60G40, 91A60]

Bernoulli bandit process Markov decision process multi-armed bandit limit rule play-the-winner rule least failures rule discount optimality


Kelly, F. P. Multi-Armed Bandits with Discount Factor Near One: The Bernoulli Case. Ann. Statist. 9 (1981), no. 5, 987--1001. doi:10.1214/aos/1176345578.

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